The chart shows the percentage of people aged 65 and over in the US
between 1900 and 2000. In the year 1900 just over 4% of the population
was aged over 65. However by 1960 this figure had doubled.
Why this sentence "However by 1960 this figure had doubled." can't be in past tense?
The Perfect, as the name suggests, is about completion.
The Past Perfect is about completion "up until sometime ago," as the present perfect is about completion "up until now."
The part "by 1960 this figure had doubled" is talking about a completion in the past. The expression 'by 1960' has the sense of duration as already commented up there, so it goes well with the perfect form.
Logically speaking, the perfect actually exists to add the sense of completion to the finite verb such as infinitives, participles, and modal verbs because these verbs are tenseless and can not talk about past events by themselves. The perfect itself is also tenseless, and thus it's vague in nature and has the sense of duration. That is why when used with a verb that has tense, it ends up saying 'up until now or sometime ago.'